### CAT 1999 Question Paper Question 38

Instructions

DIRECTIONS for the following questions: These questions are based on the situation given below:

There are m blue vessels with known volumes $$V1, V2 , ...., V_m$$, arranged in ascending order of volume, where $$v_1 > 0.5$$ litre, and $$V_m < 1$$ litre. Each of these is full of water initially. The water from each of these is emptied into a minimum number of empty white vessels, each having volume 1 litre. The water from a blue vessel is not emptied into a white vessel unless the white vessel has enough empty volume to hold all the water of the blue vessel. The number of white vessels required to empty all the blue vessels according to the above rules was n.

Question 38

# Among the four values given below, which is the least upper bound on e, where e is the total empty volume in the m white vessels at the end of the above process?

Solution

$$v_1 > 0.5$$

$$v_m < 1$$

As we cannot empty a blue vessel into a white vessel unless there is enough space to hold all the volume in blue vessel in the white vessel, we have to use m white vessels. This is because we cannot empty more than 1 blue vessel into 1 white vessel.

The minimum volume that can be left in each white vessel is $$1-v_m$$.

=> The minimum volume that can be left in m white vessels is $$m(1-v_m)$$, which is the least upper bound.